Answer: b) 23.94
<u>Step-by-step explanation:</u>
You are trying to find the area under the curve. Area = height x width.
Height is the y-value at the given coordinate --> f(x)
Width is the distance between the x-values --> dx
First, let's figure out dx: the distance from 1 to 4 is 3 units. We need to divide that into 6 sections because n = 6 --> dx = 1/2
So the intervals are when x = {1, 3/2, 2, 5/2, 3, 7/2, 4}
For midpoint sum, we need to find the midpoint of the intervals
![\text{midpoint}\ \bigg\{1, \dfrac{3}{2}\bigg\}=\dfrac{1+\frac{3}{2}}{2}=\dfrac{5}{4}\\\\\\\text{midpoint}\ \bigg\{\dfrac{3}{2}, 2\bigg\}=\dfrac{\frac{3}{2}+2}{2}=\dfrac{7}{4}\\\\\\\text{midpoint}\ \bigg\{2, \dfrac{5}{2}\bigg\}=\dfrac{2+\frac{5}{2}}{2}=\dfrac{9}{4}\\\\\\\text{midpoint}\ \{\dfrac{5}{2}+3\}=\dfrac{\frac{5}{2}+3}{2}=\dfrac{11}{4}\\\\\\\text{midpoint}\ \{3, \dfrac{7}{2}\}=\dfrac{3+\frac{7}{2}}{2}=\dfrac{13}{4}\\\\\\\text{midpoint}\ \{\dfrac{7}{2},4\}=\dfrac{\frac{7}{2}+4}{2}=\dfrac{15}{4}](https://tex.z-dn.net/?f=%5Ctext%7Bmidpoint%7D%5C%20%20%5Cbigg%5C%7B1%2C%20%5Cdfrac%7B3%7D%7B2%7D%5Cbigg%5C%7D%3D%5Cdfrac%7B1%2B%5Cfrac%7B3%7D%7B2%7D%7D%7B2%7D%3D%5Cdfrac%7B5%7D%7B4%7D%5C%5C%5C%5C%5C%5C%5Ctext%7Bmidpoint%7D%5C%20%20%5Cbigg%5C%7B%5Cdfrac%7B3%7D%7B2%7D%2C%202%5Cbigg%5C%7D%3D%5Cdfrac%7B%5Cfrac%7B3%7D%7B2%7D%2B2%7D%7B2%7D%3D%5Cdfrac%7B7%7D%7B4%7D%5C%5C%5C%5C%5C%5C%5Ctext%7Bmidpoint%7D%5C%20%20%5Cbigg%5C%7B2%2C%20%5Cdfrac%7B5%7D%7B2%7D%5Cbigg%5C%7D%3D%5Cdfrac%7B2%2B%5Cfrac%7B5%7D%7B2%7D%7D%7B2%7D%3D%5Cdfrac%7B9%7D%7B4%7D%5C%5C%5C%5C%5C%5C%5Ctext%7Bmidpoint%7D%5C%20%20%5C%7B%5Cdfrac%7B5%7D%7B2%7D%2B3%5C%7D%3D%5Cdfrac%7B%5Cfrac%7B5%7D%7B2%7D%2B3%7D%7B2%7D%3D%5Cdfrac%7B11%7D%7B4%7D%5C%5C%5C%5C%5C%5C%5Ctext%7Bmidpoint%7D%5C%20%20%5C%7B3%2C%20%5Cdfrac%7B7%7D%7B2%7D%5C%7D%3D%5Cdfrac%7B3%2B%5Cfrac%7B7%7D%7B2%7D%7D%7B2%7D%3D%5Cdfrac%7B13%7D%7B4%7D%5C%5C%5C%5C%5C%5C%5Ctext%7Bmidpoint%7D%5C%20%20%5C%7B%5Cdfrac%7B7%7D%7B2%7D%2C4%5C%7D%3D%5Cdfrac%7B%5Cfrac%7B7%7D%7B2%7D%2B4%7D%7B2%7D%3D%5Cdfrac%7B15%7D%7B4%7D)
Next, let's find the height for each of the midpoints:
f(x) = x² + 1
![f\bigg(\dfrac{5}{4}\bigg)=\bigg(\dfrac{5}{4}\bigg)^2+1=\dfrac{41}{6}\\\\\\f\bigg(\dfrac{7}{4}\bigg)=\bigg(\dfrac{7}{4}\bigg)^2+1=\dfrac{65}{6}\\\\\\f\bigg(\dfrac{9}{4}\bigg)=\bigg(\dfrac{9}{4}\bigg)^2+1=\dfrac{97}{6}\\\\\\f\bigg(\dfrac{11}{4}\bigg)=\bigg(\dfrac{11}{4}\bigg)^2+1=\dfrac{137}{6}\\\\\\f\bigg(\dfrac{13}{4}\bigg)=\bigg(\dfrac{13}{4}\bigg)^2+1=\dfrac{185}{6}\\\\\\f\bigg(\dfrac{15}{4}\bigg)=\bigg(\dfrac{15}{4}\bigg)^2+1=\dfrac{241}{6}](https://tex.z-dn.net/?f=f%5Cbigg%28%5Cdfrac%7B5%7D%7B4%7D%5Cbigg%29%3D%5Cbigg%28%5Cdfrac%7B5%7D%7B4%7D%5Cbigg%29%5E2%2B1%3D%5Cdfrac%7B41%7D%7B6%7D%5C%5C%5C%5C%5C%5Cf%5Cbigg%28%5Cdfrac%7B7%7D%7B4%7D%5Cbigg%29%3D%5Cbigg%28%5Cdfrac%7B7%7D%7B4%7D%5Cbigg%29%5E2%2B1%3D%5Cdfrac%7B65%7D%7B6%7D%5C%5C%5C%5C%5C%5Cf%5Cbigg%28%5Cdfrac%7B9%7D%7B4%7D%5Cbigg%29%3D%5Cbigg%28%5Cdfrac%7B9%7D%7B4%7D%5Cbigg%29%5E2%2B1%3D%5Cdfrac%7B97%7D%7B6%7D%5C%5C%5C%5C%5C%5Cf%5Cbigg%28%5Cdfrac%7B11%7D%7B4%7D%5Cbigg%29%3D%5Cbigg%28%5Cdfrac%7B11%7D%7B4%7D%5Cbigg%29%5E2%2B1%3D%5Cdfrac%7B137%7D%7B6%7D%5C%5C%5C%5C%5C%5Cf%5Cbigg%28%5Cdfrac%7B13%7D%7B4%7D%5Cbigg%29%3D%5Cbigg%28%5Cdfrac%7B13%7D%7B4%7D%5Cbigg%29%5E2%2B1%3D%5Cdfrac%7B185%7D%7B6%7D%5C%5C%5C%5C%5C%5Cf%5Cbigg%28%5Cdfrac%7B15%7D%7B4%7D%5Cbigg%29%3D%5Cbigg%28%5Cdfrac%7B15%7D%7B4%7D%5Cbigg%29%5E2%2B1%3D%5Cdfrac%7B241%7D%7B6%7D)
Now, let's find the Area: A = f(x) dx:
![\text{Midpoint Sum:}\quad A=\dfrac{1}{2}\bigg(\dfrac{41}{16}+\dfrac{65}{16}+\dfrac{97}{16}+\dfrac{137}{16}+\dfrac{185}{16}+\dfrac{241}{6}\bigg)\\\\\\.\qquad \qquad \qquad \qquad =\dfrac{1}{2}\bigg(\dfrac{766}{16}\bigg)\\\\\\.\qquad \qquad \qquad \qquad =\dfrac{383}{16}\\\\\\.\qquad \qquad \qquad \qquad =23.9375](https://tex.z-dn.net/?f=%5Ctext%7BMidpoint%20Sum%3A%7D%5Cquad%20A%3D%5Cdfrac%7B1%7D%7B2%7D%5Cbigg%28%5Cdfrac%7B41%7D%7B16%7D%2B%5Cdfrac%7B65%7D%7B16%7D%2B%5Cdfrac%7B97%7D%7B16%7D%2B%5Cdfrac%7B137%7D%7B16%7D%2B%5Cdfrac%7B185%7D%7B16%7D%2B%5Cdfrac%7B241%7D%7B6%7D%5Cbigg%29%5C%5C%5C%5C%5C%5C.%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%3D%5Cdfrac%7B1%7D%7B2%7D%5Cbigg%28%5Cdfrac%7B766%7D%7B16%7D%5Cbigg%29%5C%5C%5C%5C%5C%5C.%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%3D%5Cdfrac%7B383%7D%7B16%7D%5C%5C%5C%5C%5C%5C.%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%3D23.9375)